Two derivative problems

So, my homework problem is deriving -2*sin(x)*cos(x) as part of using the concavity theorem on -6x^2+cos^2(x). My calculator is saying 2-4(cos(x))^2, but I'm not getting to the same answer.

My other question is working the problem sin^2(cos(4x)), this was a problem on my exam that I got right, but only because I memorized the general pattern of the problem when we got it in class the week before and she just happened to have it on the test. I need to see it worked out step by step, as I get lost in it pretty quickly.

So, my homework problem is deriving -2*sin(x)*cos(x) as part of using the concavity theorem on -6x^2+cos^2(x). My calculator is saying 2-4(cos(x))^2, but I'm not getting to the same answer.

My other question is working the problem sin^2(cos(4x)), this was a problem on my exam that I got right, but only because I memorized the general pattern of the problem when we got it in class the week before and she just happened to have it on the test. I need to see it worked out step by step, as I get lost in it pretty quickly.

So, my homework problem is deriving -2*sin(x)*cos(x) as part of using the concavity theorem on -6x^2+cos^2(x). My calculator is saying 2-4(cos(x))^2, but I'm not getting to the same answer.

My other question is working the problem sin^2(cos(4x)), this was a problem on my exam that I got right, but only because I memorized the general pattern of the problem when we got it in class the week before and she just happened to have it on the test. I need to see it worked out step by step, as I get lost in it pretty quickly.

( It works out to -8*sin(4x)*sin(cos(4x)) * cos(cos(4x)) )

Thanks!

For your 2nd question (which hasn't been answered):

Use:

The Chain Rule

Originally Posted by Wolvenmoon

I memorized the general pattern of the problem

If this is the "general pattern" you memorized, you did the right thing.

In this case you want to differentiate

Using the chain rule:

Note that f(x) is itself a combo function with the trapped inside the

So the derivative of is

A trig identity says that

So

Notice the g(x) is also a combo with the 4x trapped inside the .

So the derivative of is

Now using the master chain rule:

The derivative of is

Substitute g'(x) and substitute g(x) for x in the f'(x) calculation:

Phew! And that's the answer.

NOTE: This answer is a bit more simplified that the one you got because I used the trig identity . You can check that they are both the same function, by graphing them or randomly checking points such as x=3. I have checked and saw that they are the same, so my answer is also correct.